Skip to main content
Log in

A Real Method for Solving Quaternion Matrix Equation \(\mathbf{X}-\mathbf{A}{\hat{\mathbf{X}}}{} \mathbf{B} = \mathbf{C}\) Based on Semi-Tensor Product of Matrices

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

In this paper, we study the quaternion matrix equation (1.1) by using the semi-tensor product of matrices. First, we propose a real vector representation of a quaternion matrix and study its properties. Then combining this real vector representation with semi-tensor product of matrices, the explicit expressions of the least squares solution, the least squares J-self-conjugate solution and the least squares anti-J-self-conjugate solution of (1.1) are proposed. We give the equivalence conditions for the compatibility of (1.1) and the expressions of the minimal norm solution with above-mentioned properties. We also propose the corresponding algorithms and give numerical experiments to verify the effectiveness of our methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Adler, S.L.: Scattering and decay theory for quaternionic quantum mechanics and the structure of induced nonconservation. Phys. Rev. D. 37(12), 3654–3662 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  2. Bihan, N.L., Sangwine, S.J.: Color image decomposition using quaternion singular value decomposition. In: Proceedings of IEEE International Conference on Visual Information Engineering of Quaternion, pp. 985–998. VIE, Guidford (2003)

  3. Caccavale, F., Natale, C., Siciliano, B., Villani, L.: Six-dof impedance control based on angle/axis representations. IEEE Trans. Robot. Autom. 15(2), 289–300 (1999)

    Article  Google Scholar 

  4. Cheng, D.Z., Feng, J.E., Lv, H.: Solving fuzzy relational equations via semi-tensor product. IEEE. Trans. Fuzzy. Syst. 20(2), 390–396 (2012)

    Article  Google Scholar 

  5. Cheng, D.Z., Qi, H.S., He, F.H.: Mappings and Dynamic systems over Finite sets-A Semi-tensor Product Approach. Science Press, Beijing (2016)

    Google Scholar 

  6. Cheng, D.Z., Qi, H.S., Li, Z.: Stability and stabilization of Boolean networks. Int. J. Robust. Nonlin. 21, 134–156 (2011)

    Article  MathSciNet  Google Scholar 

  7. Cheng, D.Z., Qi, H.S., Li, Z.Q.: Analysis and Control of Boolean Net-Works: A Semi-Tensor Product Approach. Springer, London (2011)

    Book  Google Scholar 

  8. Cheng, D.Z., Qi, H.S., Xue, A.C.: A survey on semi-tensor product of matrices. J. Syst. Sci. Complex. 20, 304–322 (2007)

    Article  MathSciNet  Google Scholar 

  9. Cheng, D.Z., Qi, H.S., Zhao, Y.: An Introduction to Semi-Tensor Product of Matrices and its Application. World Scientific Publishing Company, Singapore (2012)

    Book  Google Scholar 

  10. Davies, A.J.: Quaternionic Dirac equation. Phys. Rev. D. 41, 2628–2630 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  11. Farenick, D.R., Pidkowich, B.A.F.: The spectral theorem in quaternions. Linear Algebra Appl. 371, 75–102 (2003)

    Article  MathSciNet  Google Scholar 

  12. Ghouti, L.: Robust perceptual color image hashing using quaternion singular value decomposition. In: IEEE International Conference on Acoustics. IEEE (2014)

  13. Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. The Johns Hopkins University Press, Baltimore (2013)

    MATH  Google Scholar 

  14. Ji, P., Wu, H.T.: A closed-form forward kinematics solution for the 6–6p stewart platform. IEEE Trans. Robot. Autom. 17, 522–526 (2002)

    Google Scholar 

  15. Jiang, T.S., Wei, M.S.: On a solution of the quaternion matrix equation \(\mathit{X-AXB}=\mathit{C}\) and its application. Acta. Math. Sin. 21(3), 483–490 (2005)

    Article  MathSciNet  Google Scholar 

  16. Moxey, C.E., Sangwine, S.J., Ell, T.A.: Hypercomplex correlation techniques for vector imagines. IEEE Trans. Signal Process. 51, 1941–1953 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  17. Song, C., Chen, G.: On solutions of the matrix equation \(\mathit{XF}-\mathit{AX}=\mathit{C}\) and \(\mathit{XF}-\mathit{A}{\widetilde{X}}=\mathit{C}\) over quaternion field. J. Appl. Math. Comput. 37(1–2), 57–68 (2011)

    Article  MathSciNet  Google Scholar 

  18. Song, C.Q., Chen, G.L., Wang, X.D.: On solutions of quaternion matrix equation \(\mathit{XF}-\mathit{A}{\widetilde{X}}=\mathit{BY}\) and \(\mathit{XF}-\mathit{AX}=\mathit{BY}\). Acta. Math. Sci. 32(5), 1967–1982 (2012)

    Article  MathSciNet  Google Scholar 

  19. Song, C.Q., Feng, J.E., Wang, X.D., Zhao, J.L.: A real representation method for solving yakubovich-j-conjugate quaternion matrix equation. Abstr. Appl. Anal. 5, 1–9 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Wei, M.H., Wei, M.S., Feng, Y.: An iterative algorithm for least squares problem in quaternionic quantum theory. Comput. Phys. Commun. 179(4), 203–207 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  21. Wei, M.S., Li, Y., Zhang, F.X., Zhao, J.L.: Quaternion Matrix Computations. Nova Science Publisher, New York (2018)

    Google Scholar 

  22. Xu, M., Wang, Y., Wei, A.: Robust graph coloring based on the matrix semi-tensor product with application to examination timetabling. Control Theory Technol. 12(2), 187–197 (2014)

    Article  MathSciNet  Google Scholar 

  23. Xu, X., Hong, Y.: Matrix approach to model matching of asynchronous sequential machines. IEEE. Trans. Autom. Control. 58(11), 2974–2979 (2013)

    Article  MathSciNet  Google Scholar 

  24. Yuan, S.F., Liao, A.P.: Least squares solution of the quaternion matrix equation \(\mathit{X}-\mathit{A}{\hat{X}}\mathit{B}=\mathit{C}\) with the least norm. Linear Multilinear Algebra 59(9), 113–116 (2011)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ying Li.

Additional information

Communicated by Dietmar Hildenbrand.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

W. Ding: This work was completed with the support of the Natural Science Foundation of Shandong under grant ZR2020MA053.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ding, W., Li, Y. & Wang, D. A Real Method for Solving Quaternion Matrix Equation \(\mathbf{X}-\mathbf{A}{\hat{\mathbf{X}}}{} \mathbf{B} = \mathbf{C}\) Based on Semi-Tensor Product of Matrices. Adv. Appl. Clifford Algebras 31, 78 (2021). https://doi.org/10.1007/s00006-021-01180-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00006-021-01180-1

Keywords

Mathematics Subject Classification

Navigation